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A review of the recent research on vibration energy harvesting via bistable systems

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2013 Smart Mater. Struct. 22 023001

(http://iopscience.iop.org/0964-1726/22/2/023001)

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IOP PUBLISHING SMARTMATERIALS ANDSTRUCTURES

Smart Mater. Struct. 22 (2013) 023001 (12pp) doi:10.1088/0964-1726/22/2/023001

TOPICAL REVIEW

A review of the recent research onvibration energy harvesting via bistablesystems

R L Harne and K W Wang

Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125, USA

E-mail:[email protected]

Received 25 July 2012, in final form 9 December 2012

Published 25 January 2013

Online atstacks.iop.org/SMS/22/023001

Abstract

The investigation of the conversion of vibrational energy into electrical power has become a

major field of research. In recent years, bistable energy harvesting devices have attracted

significant attention due to some of their unique features. Through a snap-through action,

bistable systems transition from one stable state to the other, which could cause large

amplitude motion and dramatically increase power generation. Due to their nonlinear

characteristics, such devices may be effective across a broad-frequency bandwidth.

Consequently, a rapid engagement of research has been undertaken to understand bistableelectromechanical dynamics and to utilize the insight for the development of improved

designs. This paper reviews, consolidates, and reports on the major efforts and findings

documented in the literature. A common analytical framework for bistable electromechanical

dynamics is presented, the principal results are provided, the wide variety of bistable energy

harvesters are described, and some remaining challenges and proposed solutions are

summarized.

(Some figures may appear in colour only in the online journal)

1. Introduction

Vibrational energy harvesting studies have begun adopting

the perspective that linear assumptions and stationary

excitation characteristics used in earlier analyses and designs

are insufficient for the application of harvesters in many

realistic environments. The principal challenge is that

linear oscillators, well suited for stationary and narrowband

excitation near their natural frequencies, are less efficient

when the ambient vibrational energy is distributed over a wide

spectrum, may change in spectral density over time, and is

dominant at very low frequencies [1,2].

These factors encouraged the exploration of methods to

broaden the usable bandwidth of linear harvesters, including

oscillator arrays, multi-modal oscillators, and active oradaptive frequency-tuning methods [3, 4]. While providing

improvements, more advanced solutions were desired for

broadband performance, and the exploitation of nonlinearitybecame a subsequent focus. To date, a number of nonlinear

energy harvesting studies have been conducted, mostly

focusing on the monostable Duffing [57], impact [8,

9], and bistable oscillator designs. Monostable Duffing

harvesters exhibit a broadening resonance effect dependent

on the nonlinearity strength, device damping, and excitation

amplitude, and thus can widen the usable bandwidth of

effective operation. Impact harvesters provide a mechanism

for frequency up-conversion by using lower ambient vibration

frequencies to impulsively excite otherwise linear harvesters

so that they may ring down from much higher natural

frequencies.

Bistable oscillators have a unique double-well restoringforce potential, as depicted in figure1.This provides for three

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Smart Mater. Struct. 22 (2013) 023001 Topical Review

Figure 1. Double-well restoring force potential of a bistableoscillator showing example trajectories for (a) intrawell oscillations,

(b) chaotic interwell vibrations and (c) interwell oscillations.

distinct dynamic operating regimes depending on the input

amplitude, figure2. Bistable devices may exhibit low-energy

intrawell vibrations (figure1(a)). In this case, the inertial massoscillates around one of the stable equilibria with a small

stroke per forcing period; see the example displacementtime

response trajectory (figure 2(a)) and phase portrait with an

overlay Poincare map (figure2(d)). Alternatively, the bistable

oscillator may be excited to a degree so as to exhibit

aperiodic or chaotic vibrations between wells (figures 1(b),2(b) and (e)). As the excitation amplitude is increased still

further, the device may exhibit periodic interwell oscillations(figures 1(c), 2(c) and (f)). In some cases, the dynamic regimes

may theoretically coexist although only one is physically

realizable at a time.

The periodic interwell vibrationsalternatively, high-

energy orbits or snap-throughhave been recognized as a

means by which to dramatically improve energy harvesting

performance [3, 4]. As the inertial mass must displace

a greater distance from one stable state to the next, therequisite velocity of the mass is much greater than that for

intrawell or chaotic vibrations. Since the electrical output

of an energy harvester is dependent on the mass velocity,

high-energy orbits substantially increase power per forcing

cycle (as compared with intrawell and chaotic oscillations)

and are more regular in waveform (as compared with

chaotic oscillations), which is preferable for external power

storage circuits. Additionally, snap-through may be triggered

regardless of the form or frequency of exciting vibration,

alleviating concerns about harvesting performance in many

realistic vibratory environments dominated by effectively

low-pass filtered excitation[10].

These benefits have instigated a rapidly growing body ofliterature on bistable energy harvesting. Among many, three

common bistable harvester concepts are depicted in figure3.

Harvesting circuitry is indicated by the parallelogram, and

attached piezoelectric patches for converting mechanical

strain to electrical energy are shown as light gray layers

partially covering the beam lengths. The direction of base

excitation is indicated by the double arrows. Figure 3(a)

shows a magnetic repulsion harvester with the strength of

the nonlinearity governed by the magnet gap distance dr.

Figure 3(b) shows a magnetic attraction bistable harvester

using a ferromagnetic beam directed towards one of two

magnets separated a distance 2dgfrom each other anddafromthe end of the beam. Lastly, figure3(c) shows an example of

Figure 2. Example displacementtime responses (top row) and phase plots with an overlap Poincare map as black circles (bottom row) for

three dynamic regimes of bistable oscillators. (a) and (d) Intrawell oscillations. (b) and (e) Chaotic vibrations. (c) and (f) Interwelloscillations.

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Figure 3. (a) Bistable magnetic repulsion harvester. (b) Magneticattraction harvester. (c) Buckled beam harvester. Piezoelectricpatches shown as light gray layers along part of the beam lengths.Harvesting circuitry shown as parallelograms.

a buckled beam harvester with the bistability modified by a

variable axial loadp. Note that while these three are used hereas examples for illustration, this review is not limited to these

types of devices.The fundamental electromechanical dynamics have been

evaluated analytically and experimentally with respect to the

individual bistable device design under consideration, but

a common dimensionless formulation is often utilized that

yields trends comparable across platforms. The advantages

of bistability in both stationary and stochastic vibratory

environments have also been detailed. Numerous studies haveprobed these subjects to various levels of refinement. As a

result, a rigorous and comprehensive review of the bistable

energy harvesting literature would be an important service tothe technical community. A previous paper has summarized

a portion of bistable energy harvesting developments to

date, though the authors impart particular emphasis to

survey studies regarding MEMS-scale utility and efficiency

metrics [11]. In contrast to the prior survey, the objective

of the present review is to provide a comprehensive outline

of the recent bistable energy harvesting literature, so as to

encompass the breadth of work accomplished and provide

sufficient attention to critical results of these studies.In the following sections, this review organizes the

variety of research investigations in bistable energy harvesting

based on similar analytical methods and experimental

conceptions. Following the presentation of a unifiedelectromechanical analytical model widely employed by

researchers, principal conclusions from analytical studies are

summarized. Thereafter, the great body of experimental work

is surveyed and additional insights observed experimentally

but not captured in fundamental analysis are highlighted.

Finally, remaining challenges to the field, proposed solutions

to these obstacles, and the relation between bistable energy

harvesting and similar explorations in contemporaneous fieldsare summarized.

2. Governing equations of single degree-of-freedombistable oscillator

The interest in bistable oscillator dynamics grew in

proportion to the discovery of the attendant chaotic

oscillations which occur for specific operating parameters,

first observed numerically and experimentally by Tseng and

Dugundji [12]. The authors described the chaotic vibrations of

a buckled beam as intermittent snap-through[12]. Extensive

exploration was performed later by Holmes [13] and Moon

and Holmes [14] so that a more detailed understanding

developed from which the recent literature in bistable energy

harvesting has taken root. The governing equation derived was

for a mechanically buckled beam [13] and for a beam buckled

via magnetic attraction [14]. Using a one-mode Galerkin

approximation, the authors derived an ordinary differential

governing equation for the buckled beams which was found

to accurately represent experimental results.

The governing equation for an underdamped, single

degree-of-freedom oscillator excited by base acceleration may

be formulated from the physical coordinates where the relative

displacementX(t)of an inertial mass mis determined by

m X+c X+ dU(X)dX

= m Z (1)

where c is the viscous damping constant, Z is the inputbase acceleration, and the overdot denotes differentiation with

time. The restoring force potential of the oscillator may be

expressed as

U(X)= 12 k1(1r)X2 + 14 k3X4 (2)where k1 is the linear spring constant, k3 is the nonlinear

spring constant, and r is a tuning parameter. Figure 4 shows

the effect on the restoring force potential for three cases of

tuning parameter and nonlinearity strength, = k3/k1. Thelinear oscillator, = 0 and r < 1, is monostable as is thenonlinear Duffing oscillator,=0 andr1 which exhibits asoftening nonlinearity for < 0 and hardening nonlinearity

when > 0. However, when the tuning parameter r > 1

and > 0, the central equilibrium is no longer stable and

the system becomes nonlinear bistable, having new stable

equilibria at X= (r1)/. This latter case is alsoreferred to as the DuffingHolmes oscillator in honor of their

collective contributions [15].

A nondimensional time,=t, is applied to equation (2)where

=

k1/m is the linear natural frequency of

the oscillator. Defining = c/2m, and operator ()as differentiation with respect to , the nondimensional

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Figure 4. Spring force potential as tuning and nonlinearity aremodified.

governing equation is given as

x+2x+(1r)x+x3 = z. (3)Equation (3) is the Duffing equation, which has a

rich history [15]. Despite representing a purely mechanical

system, equation (3) served as the initial model for a

number of bistable energy harvesting studies [1618]. This

convention stems from early energy harvesting literature in

which the coupled external circuit was modeled as equivalent

damping [19]. While this is an incomplete perspective, it

provides fundamental insight into the role of bistability inelectromechanical dynamics and justifies its adoption.

On the other hand, coupling effects between the external

harvesting circuit and the electromechanical device have been

rigorously studied and verified, particularly as related to

harvesting efficiency[2023]. As a result, more recent studies

have included a coupled external circuit equation for greater

fidelity. Depending on both the manifestation of bistability

that is considered and the form of electromechanical coupling,

different equations of motion in physical coordinates are

attained. Drawing on a number of recent works [2427], the

authors here provide a framework which collects together

the electromechanical conversion mechanisms most often

considered: electromagnetic and piezoelectric.Figure5(a) shows a generic electromechanical oscillator

with restoring force dU/dx, base excitation Z, and piezo-electric and electromagnetic conversion mechanisms. The

subsequent external circuits connected to the piezoelectric

and electromagnetic mechanisms are depicted in figures5(b)

and (c), respectively. To adopt a common convention in the

literature, the external harvesting circuits are described by a

generic load resistance[2427]. The governing equations are

found to be

m X+c X+ dU(X)dX

+V+I= m Z (4)

CpV+ 1R1

VX= 0 (5)

LI+R2IX=0 (6)where is the linear piezoelectric coupling coefficient;V is the voltage across the load resistance R1 for the

piezoelectric harvesting component; Cp is the capacitance of

the piezoelectric material; is the electromagnetic coupling

coefficient; I is the current through the load resistance R2for the electromagnetic harvesting component, where thetotal resistance is the sum of a coil resistance and the

harvesting circuit resistance; and L is the inductance of theelectromagnetic mechanism.

Introducing new coordinates

x=X; z=Z;=CpV/; i=LI/

(7)

and employing the nondimensional time, = t, where=k1/mis again the linear natural frequency of the oscillator,

the dimensionless system of equations is determined as

x+2x+(1r)x+x3 + 2+2i= z (8)+x= 0 (9)

i+ ix= 0 (10)where the following variables are defined

2= cm

; = k3k1

; 2 = 2

k1Cp;

2 = 2

k1L; = 1

R1Cp; = R2

L.

(11)

In this notation, and are linear piezoelectric and

electromagnetic coupling coefficients, respectively; while

and are the nondimensional frequencies of the piezoelectricand electromagnetic components, respectively, normalized

relative to the linear natural frequency of the mechanical

oscillator. Bistable energy harvesting studies focus on the

case in which the tuning parameter r > 1. Furthermore,several works cited in this review define the negative linear

stiffness such that r=2, reducing the number of parametersin equation(8).

3. Analysis approaches and results

Equations (8)(10) are the foundation for many recent bistable

energy harvesting analyses. Should an individual study beconcerned with only one of the electromechanical conversion

methods, the unrelated equation and coupling components are

omitted. Although not all authors report their exact approach,a variety of analytical techniques exist to predict the response

of a bistable system governed by equation (8)(10). The type

of information produced by each method is unique and theprincipal insight obtained is characteristic to the analytics;

thus, it is natural to distinguish the key results based on the

analytical methods.

3.1. Numerical integration

Following conversion to state-space, many studies thereafterpredict system response via numerical integration. The

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Figure 5. (a) Representative mechanical schematic of bistable energy harvester having both piezoelectric and electromagnetic conversions.(b) Equivalent coupled circuit for piezoelectric element. (c) Equivalent circuit for electromagnetic element.

benefits to this approach are the relative straightforward nature

of simulation and the versatility to model arbitrary input

excitations. The main drawbacks are the computational time

and cost for the requisite repeated simulations if one is

interested in frequency-domain information and the difficulty

in attaining a deep and comprehensive insight into the systemdynamics.

Erturk and Inman [26] used numerical integration to

compare the qualitative similarity of simulated and measured

data for a bistable piezoelectric harvester. The authors found

good agreement for time-domain predictions in terms of

the bistable harvester outperforming the linear equivalent

over a broad range of frequencies. In the study, the linear

device and the bistable harvester exhibited similar linearized

natural frequencies to provide a meaningful comparison of

performance. Apart from near the linear harvesters natural

frequency, the bistable harvester consistently yielded greater

levels of output power, so long as the bistable device

maintained a high-energy orbit. In the event that chaoticoscillations were induced, the bistable device provided only

a marginal increase in RMS voltage output.Several studies have predicted system response to random

noise inputs via numerical integration [2830]. Litaket al[29]

observed that a certain level of white Gaussian input appeared

to maximize the output power of the device, which was

explained to be the result of inducing a form of stochastic

resonance. This phenomenon is the combined result of a

small periodic force acting on the oscillator, so as to create

a dynamic double-well potential, and a certain level of

input noises which collectively induce dramatic interwell

oscillations [31,32]. McInnes et al [17]proposed exploitingthis feature in energy harvesting, providing simulations in

which the triggering of stochastic resonance significantly

improved the bistable harvester performance compared to

intrawell vibrations. The study by Litak et al [29] did not

provide for a dynamic double-well potential but did observe

that an optimum level of stochastic excitation existed even for

a static restoring force potential to maximize output voltage.

A bistable plate having piezoelectric patches for energy

harvesting has been investigated [33, 34]. The system

response was assumed to be the coupled dynamics of

the two unique stable modes and a subharmonic behavior.

Rather than a continuum approach, the three coupled

responses were approximated as individual out-of-planedisplacements at a given point on the plate, thus representing

the relative contribution of the three responses to a given

excitation. Curve fitting with a quadratic polynomial was

used to characterize the nonlinear restoring force. In spite

of the simplifying assumptions, the numerically integrated

simulations agreed well with experimental data, particularly

given the discontinuous nonlinearity of the restoring force [35,36].

3.2. Harmonic balance

Harmonic balance is advantageous for providing an efficient

analytical framework to assess steady-state dynamics. The

drawbacks are inherent in the assumption that the system

response is the superposition of a number of harmonics, and

therefore the fidelity of the method is limited to the size of the

truncated series.Stantonet al[37] found that the optimum electromechan-

ical coupling strength for energy harvesting was the maximum

value that sustained high-energy orbits. The analysis was alsoused to predict optimal load impedance conditions for energy

harvesting from snap-through vibrations; this characteristic

was demonstrated experimentally as well [26]. The resultswere compared against direct numerical integration and

showed good agreement [37].

Mann et al [38] used harmonic balance to evaluate

the effect of uncertainties inherent in a realistic energy

harvesting application, e.g., device design parameters orexcitation characteristics. A bistable harvester was compared

against linear, softening monostable Duffing, and hardening

monostable Duffing designs. Although the linear device was

predicted to provide greater average output power than all ofthe nonlinear devices at the linear resonance frequency, the

95% confidence interval for the linear device was substantial,

suggesting a high susceptibility of performance to parameter

changes. In contrast, the bistable harvester exhibited the most

consistent performance, having tightly confined confidence

intervals around the average. This provides proof of the

robustness of the bistable harvester to a changing excitation

environment as well as imperfect knowledge of designparameters.

3.3. Method of multiple scales (MMS)

The method of multiple scales yields steady-state andtransient solutions under the assumption of small perturbation

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of vibrations. Karami and Inman [25] used MMS in the

analysis of the bistable energy harvester with an expansion

to three time scales. It was shown that the MMS solution

had reduced efficacy as the vibration amplitude around an

equilibrium state was increased. The successful use of MMS

was verified for small deviations around either the intrawell

and interwell vibration solutions. The method was alsoutilized to determine an equivalent damping and frequency

shift induced by energy harvesting so as to reduce analysis to

a single governing equation. Simulations using this approach

agreed well with results computed from the fully coupled

electromechanical equations. Since a number of works

in bistable energy harvesting utilize only the mechanical

governing equation (3) to approximate energy dissipated,

the equivalency provides a nontrivial and computationally

efficient correction to such analyses.

3.4. Melnikovs method

The determination of design or excitation parameters

necessary to maintain high-energy orbits is crucial to the

optimum design of a bistable energy harvester. To this end,

Melnikovs theory may be employed. Melnikovs method

stems from the study of conditions suitable for homoclinic

bifurcations which characterize the transition of a bistable

systems dynamics into the chaotic regime [15,39,40]. The

key disadvantage to the approach is its conservative estimation

of the onset of bifurcation, thus limiting its usefulness as a

design tool.

Following derivation of the mechanical governing

equation, the seminal work of Holmes [13] employed

Melnikovs theory to determine the critical nondimensionalamplitude for single-frequency sinusoidal excitation. As this

was a purely mechanical formulation, this represents the

open-circuit result for the bistable energy harvester.

Stantonet al[41] applied Melnikovs method to study the

bistable piezoelectric energy harvester. For single-frequency

excitation, it was shown that the onset of homoclinic

bifurcation was most sensitive when the driving frequency

was 0.765, and that sensitivity was not a function of system

damping. A normalized load impedance was determined

to yield the greatest electrical damping for the harvester,

thereby inhibiting interwell vibrations. The inclusion of

electromechanical coupling to Melnikovs analysis of thebistable oscillator was predicted to be so influential as

to be capable of destabilizing interwell oscillations. The

authors also demonstrated that arbitrary multi-frequency

excitation could be exploited via the theory to induce

interwell oscillations, when the individual single-frequency

excitations were insufficient for that purpose. For white

Gaussian excitation, the bistable energy harvester was

predicted to provide approximately equivalent levels of power

as compared to the linear harvester design. For exponentially

correlated noise excitation, a dramatic advantage of the

bistable harvester was demonstrated. Despite the conservative

estimates predicted by the Melnikov method, the study

provided new insights into the role of the electromechanicalcoupling in inducing (or repressing) interwell vibrations[41].

3.5. Stochastic differential equation solution

Several works have documented the advantage of bistable

energy harvesters over their linear counterparts when the

excitations are stochastic [10, 27, 42, 43]. Daqaq [10]

studied the solution to the FokkerPlanckKolmogorov (FPK)

equations in the event of white Gaussian and exponentiallycorrelated noise input. It was found that when excited by

white Gaussian noise, linear and bistable inductive harvesters

yield the same mean output power, a conclusion also verified

via Melnikovs method [41]. However, Daqaq [10] noted

that many real-world stochastic vibration sources are not

purely white and would be more accurately represented

as exponentially correlated noise. In such cases, it was

demonstrated how the double-well potential may be designed

so as to yield greater power output from the bistable harvester

than the linear device. Furthermore, optimal double-well

potential shapes could be determined for inducing interwell

vibrations for exponentially correlated noise excitation. It

was shown that such potential shapes led to correspondingoptimum escapement frequencies, similar in effect to the

Kramers rate in the study of stochastic resonance [31].

These results were also verified by corresponding numerically

integrated simulations [10].

Ando et al [42] and Ferrari et al [43] utilized the

SDE Toolbox for MATLAB[44] to simulate the mechanical

response of bistable harvesters to white Gaussian noise. Both

studies observed that the power spectral density (PSD) of the

bistable mass velocity was greater than that for the linear

sample, except at the linear device natural frequency. These

results along with the work of Daqaq [10,27]demonstrate the

robustness of bistable energy harvesters in stochastic vibrationenvironments.

3.6. Signal decomposition

When a bistable oscillator is excited at frequencies much less

than the linear natural frequency, the resulting displacement

trajectory may exhibit a combination of slow and fast time

scale oscillations, figure 6. Thus, the slow excitation forces

the oscillator to jump across the double-well potential,

where it rings down at its linear natural frequency before

the input excitation forces it back across the double well:

a frequency up-conversion technique. Cohen et al [45]used slowfast decomposition to assess the dynamics of

the system for this type of excitation. The approach was

shown to accurately predict the force applied to a bistable

harvester to induce interwell escape and thereafter serve as an

impulsive excitation. Comparable experiments were carried

out to validate the analytical approach. The decomposition

technique was shown to provide a means by which to

characterize the effectiveness in frequency up-conversion

for the bistable harvester and serves as a tool for design

optimization. In addition, the authors adapted the double-well

potential used for simulation so as to be asymmetric for better

comparison with experimental data. This represents one of the

few attempts in literature to date to address asymmetry in thebistable harvester restoring force potential.

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Figure 6. Steady-state solution, computed by numerical integration, of equation(3)forz=fcos( ), where is the ratio of thedriving frequency to the linear natural frequency . Parameters:r=2, =1, = 0.1,f= 0.5, =0.1.

4. Bistable device designs and experimental studies

The above analytical investigations employed similar math-

ematical models, generally of the form as presented in

equation (8)(10). However, the intended physical embodi-

ment for each application varied from study to study. This

section summarizes the various manifestations of bistable

energy harvesters that have been designed and experimentally

investigated, categorized by their bistability mechanisms.

4.1. Magnetic attraction bistability

The use of magnetic attraction to induce the bistability

of a cantilevered ferromagnetic beam was one of thefirst investigations employed in studying the aperiodic

chaotic response of an otherwise deterministic mechanical

system [14]. It was this construction with an additional

piezoelectric patch for energy harvesting, as in figure 3(b),

which was explored by Erturk and Inman [26] for single-

frequency excitation. An order of magnitude increase in power

was generated for the bistable device, except at the linear

harvester natural frequency, in which case the comparable

linear device provided greater power. Chaotic oscillations

were not sufficient to yield substantially greater RMS voltage

than the linear device. Exceptional agreement was also

observed between simulated and measured strange attractorsof the bistable harvester[46].

Galchev et al[47] exploited impulsive snap-through as a

frequency up-conversion technique to excite linear harvesting

devices. In their configuration, a centrally suspended magnet

is attracted by two end-suspended magnets along the axis

of a tube. As base excitation increases, the central magnet

is attracted so as to magnetically attach to one of the

end magnets. The continued sinusoidal excitation thereafter

causes the release of the central magnet from one end, which

allows the end magnet to ring down through the axis of a

coil, thus inducing flow of current in a harvesting circuit.

The central magnet then snaps over and connects to the

opposite end magnet. Upon the magnets releasing due toanother half-cycle of input vibration, the end magnet rings

down through a coil, while the central magnet snaps back

to the opposite end magnet and the cycle repeats. Since

the dynamics of operation are more affected by the input

excitation amplitude than by the frequency, a unique measure

of efficiency was proposed to show the advantage of the

device in achieving broadband energy harvesting [47].

4.2. Magnetic repulsion bistability

There are numerous studies that have investigated bistable

energy harvesters using magnetic repulsion to destabilize

the linear equilibrium position [4854]. Several of these

investigations have considered a cantilevered piezoelectric

beam with magnetic tip mass, having the same polarity asa facing magnet which may be moved a certain distance to

the beam end so as to tailor the strength of the bistability

(figure 3(a)). One feature of this configuration is that the

repulsive magnets may be moved a great distance away so

as to remove the nonlinearity and provide for the comparison

against an equivalent linear harvester.

Lin and Alphenaar [50] showed that the bistable harvester

design of figure3(a) consistently yielded greater peak voltage

than the equivalent linear device when excited by pink noise.

The study utilized a rectifying circuit to compare the voltage

measured on a storage capacitor. It was observed that the

bistable harvester provided 50% greater voltage than the lineardevice.

Tang et al [53] also studied the bistable piezoelectric

beam with magnetic repulsion. An optimum magnetic

repulsion gap was observed, at which a considerable increase

in broadband power could be harvested. The voltage in a

storage capacitor was also approximately 50% greater than

that of the linear harvester when the systems were excited by

low-pass filtered stochastic vibration.

In a different experimental configuration than the prior,

Tanget al[53] used repulsive magnets to induce a ring-down

behavior from low input frequencies representative of wave

heaves. In this investigation, the piezoelectric beam having a

magnet tip mass remains stationary while a repulsive magnet(connected to the slow-heaving vibration source) passes near

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the beam tip, thus destabilizing the equilibrium position

of the beam and causing it to vibrate as if struck by an

impulse. It was observed that when the repulsive magnets

were configured so as to pass closely by each other, the

design was insensitive to changes in input frequency for power

harvesting.

Sneller et al [48] and Mann and Owens [49] usedmagnetic repulsion of a magnet oscillating along the axis of

a tube to create bistability; induction of the oscillating magnet

through a surrounding coil served as the electromechanical

conversion mechanism. Inducement of high-energy orbits was

shown to provide substantial improvement in output power. A

similar device exists in the literature without the destabilizing

mechanism [5], but no direct performance comparison was

made to show the advantage of the bistable harvester to the

monostable Duffing oscillator counterpart.

Karami et al [52] employed a circular array of

cantilevered piezoelectric beams with magnetic tip masses

that were activated via a vertical-axis windmill having a

shaft at the center of the beam array. Repulsive magnetswere connected to the windmill shaft. Thus, as the windmill

rotated, the base-fixed piezoelectric beams were excited by

the repulsion of the tip magnets and the revolving magnets.

This concept is unique compared to the other studies in this

review since mechanical vibration is not the input excitation

mechanism. An optimum angular velocity of the windmill was

observed which most excited the array of piezoelectric beams;

the optimum rate was found also to be a function of the gap

between the repulsing magnets. The dynamics of the system

were found to be highly complex in regard to the magnetic

repelling force per rotation of the windmill. Advantages of the

proposed device as compared to other piezoelectric windmillsfound in the literature [55,56]are the lowered required wind

speed to start up and the broad range of wind speeds useful

for power harvesting.

4.3. Mechanical bistability

Mechanical design and loading offer a variety of means by

which to induce bistability into energy harvesters, including

methods inspired by biological structures [84]. A readily

adjustable bistability mechanism is a clampedclamped beam

buckled by an axial load. The post-buckled beam therefore

snaps from one stable state to the other when excitedby enough input excitation. To make this concept useful

for energy harvesting, piezoelectric patches are applied to

the beam such that oscillations of the beam will strain

the piezoelectric layers, as depicted in figure 3(c). This

configuration was earlier proposed by Baker et al[57], where

experiments of frequency-swept excitation were conducted

to validate the hypothesis that the bistability could achieve

greater levels of broadband power than the same beam without

a destabilizing axial load.

Cottone et al [30] compared this bistable harvester

design with the unbuckled configuration when excited by

exponentially correlated noise. The output RMS voltage was

increased by an order of magnitude for the bistable deviceas compared with the unbuckled sample. Experiments and

numerical modeling showed an optimum input acceleration

level exists for the bistable harvester; this finding contrasts

with linear harvesters, for which increases in input

acceleration proportionally increase the harvested power.

Masana and Daqaq [5860] have carried out detailed

studies of the post-buckled piezoelectric beam. The depth of

the double-well potential was found to play a crucial role inthe benefit of the bistable harvester. The weaker the bistability

(that is, maintaining an axial load close to the critical buckling

load), the less advantage would be attained as compared

with the unbuckled beam since the restoring force potentials

were not substantially different. However, the advantage of

the bistable device over the linear device was not uniform,

with the exception at very low frequencies when the bistable

harvester was excited into high-energy orbits but the linear

harvester was weakly excited. Superharmonic dynamics were

specifically considered in a series of comparable tests and

simulations [60]. This uniquely nonlinear dynamic regime

was found to provide a substantial increase in output power

as compared to the linear harvester, so long as the device

maintained the high-energy orbit and did not degenerate into

a coexisting low-energy stable state.

The bistability of a plate may be generated by composite

laminate lay-up. The variation in ply orientation and geometry

allow for a unique tailoring of the two stable equilibria

natural frequencies. Additionally, the spread and distribution

of the piezoelectric patches on the plate surfaces may serve

as optimization parameters for energy harvesting. Following

initial modeling analyses and experimental studies to illustrate

the potential of the bistable harvester plate [33, 34], Betts

et al [61] determined optimal lay-up configurations and

aspect ratios for energy harvesting. It was found that squareplates were the optimal lamina shape despite the greater

out-of-plane deflections attainable by higher aspect ratios.

This was attributed to the unbiased nature of the square shape

in vibrating between the two stable states, whereas plates with

aspect ratios= 1 exhibit a preference to one of the stablemodes that inhibits snap-through.

Bistability induced by an applied axial load can alter-

natively be achieved using an inverted clamped piezoelectric

beam and a tip mass selected so as to buckle the system. This

configuration was extensively explored by Friswell et al[62],

who demonstrated the advantages of the design for extremely

low-frequency vibration environments. The inverted beamconfiguration was not easily excited to interwell oscillation in

experiment. As such, designing the tip mass so that the beam

was subjected to a near-critical buckling load produced the

most favorable results.

Jung and Yun [63,64]studied frequency up-conversion

methods that exploit the impulsive snap-through behavior of

a buckled beam. An array of linear cantilevered piezoelectric

harvesters was attached to a post-buckled clampedclamped

beam. Tests showed that very low frequency excitation

(one order of magnitude less in frequency than the natural

frequency of the attached linear cantilevers) was sufficient to

yield consistent power output before the ring down decayed

substantially. An optimum excitation frequency was measuredand found to be approximately one-third of the natural

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frequency of the attached cantilevers. Power harvesting

around this excitation frequency was found to be much less

sensitive to frequency changes as compared with the single

cantilevers excited at their natural frequency; this indicates

another advantage of bistable energy harvesting in providing

a broad harvesting bandwidth.

5. Challenges in bistable energy harvesting

Despite the documented potential and advantages of bistable

harvesters, it has been shown that there is room for

improvement and advancement for these nonlinear devices.

This section summarizes several of the key remaining

challenges and some proposed solutions.

5.1. Maintaining high-energy orbits

One principle challenge is the appropriate means by which to

maintain high-energy orbits for maximum power harvestingperformance. Erturk and Inman [26] demonstrated that a

mechanical shock to the system could help the bistable

harvester recover a high-energy orbit when it was earlier

in intrawell or chaotic vibration. Masuda and Senda [65]

observed that a sudden change in external circuit impedance

could destabilize the intrawell vibration, returning the

oscillator into a high-energy orbit. Sebald et al[66] described

a similar technique whereby an impulsive voltage could be

applied in the harvesting circuit to achieve the same objective.

These methods are external interventions which require some

form of monitoring and activation. As a result, the benefit

of the approaches must be evaluated by how much energy is

expended relative to the overall harvested power.Understanding of the excitation characteristics required

to induce interwell dynamics is an area of rigorous

mathematical investigation. Melnikov theory [41], period-

doubling bifurcation [67], and evaluation of Lyapunov

exponents [40, 68] are all candidate efforts to quantify the

threshold between intrawell and interwell oscillations. Since

sustaining high-energy orbits is critical to maximizing energy

harvesting performance, and interventionary measures as

mentioned above reduce the net output, a clear knowledge

of what design and operational parameters are necessary to

maintain high-energy orbits is required. Further analytical

investigation and subsequent experimental validation arestill necessary to better characterize the sustainability of

high-energy bistable dynamics.

5.2. Operation in a stochastic vibratory environment

Realistic vibration environments for which energy harvesters

are employed are likely composed of multi-frequency

harmonics as well as a substantial proportion of low-pass

filtered noise. Although it has been shown that a bistable

device may outperform the linear equivalent in stochastic

environments, this conclusion draws on the assumption that

the bistable harvester exhibits interwell vibrations. Should

intrawell vibrations be observed, it has been proposed toutilize the random excitation component in tandem with

small coherent sinusoidal excitation to induce stochastic

resonance [31, 32]. McInnes et al [17] demonstrated that

this combination could be successfully exploited to induce

interwell oscillations in a bistable harvester. After subtracting

the theoretical active input power to modify the oscillator

potential, the net power harvested was substantially greater

than that harvested from passive intrawell vibrations. Litaket al [29] showed that a specific noise intensity maximizes

the harvested power from bistable devices having a static

potential-energy profile. Thus, in a realistic environment

where the designer knows a typical stochastic vibration

strength will dominate, the bistable harvester could be

optimally designed. This conclusion was also verified

analytically[10].

Chaotic oscillations of the bistable harvester are

preferable to intrawell vibrations, but attaining high-energy

orbits is the optimal goal. However, stochastic input

vibrations in many environments may not contain the correct

harmonics so as to sustain primarily periodic high-energy

orbits; thus, aperiodic response may dominate a bistableenergy harvesters behavior. The difficulty in harvesting

a chaotic or aperiodic output voltage as useful electrical

power has been recognized [26,37], although contemporary

work has provided some solutions with optimal stochastic

energy harvesting controls [69]. The reality of ambient

environmental vibration as compared to an ideal, stationary,

and sinusoidal input makes for the ultimate challenge in

practical energy harvesting. Fortunately, one of the advantages

of bistable energy harvesters is their robustness to real-world

unknowns [38]. At present, many of the initial investigations

in stochastic energy harvesting encourage continued study

and, in particular, experimental validation.

5.3. Coupled bistable harvesters

There has long been interest in the study of coupled systems

exhibiting chaotic behavior for the means of advantageous

synchronization and array control [70]. A recent study

evaluated the dynamics of coupled underdamped bistable

oscillators [71]. It was observed that stochastic resonance

could be induced only with moderate damping regardless

of coupling strength. However, optimal coupling and noise

strength parameters could be determined which would yield

greater signal-to-noise ratio than the uncoupled oscillators.While no decisive conclusions were offered to characterize the

complex coupling dynamics, these findings show potential for

the achievement of stochastic resonance in coupled harvesting

systems.

To the authors knowledge, only an initial study by Litak

et al [72] has thus far considered the possibility of coupling

bistable harvesters. In this report, a numerical investigation

was carried out for swept single-frequency excitation with the

harvesters coupled through a collective circuit. It was shown

that identically excited bistable devices having different

linear resonances could become unsynchronized, leading one

harvester to vibrate chaotically where it would otherwise

vibrate in high-energy orbits when uncoupled. While a singlebistable oscillator exhibits an organized Poincare map when

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undergoing chaotic vibrations, see figure 2(e), the authors

observed by case study that coupling may break down such

a Poincare map structure. The mathematical challenges posed

in the study of coupled bistable systems show this is still an

area in need of investigation.

5.4. Performance metrics

There is no standing consensus in the literature on the

preferred means by which to evaluate energy harvesting

performance [11, 21, 22] since the type of input excitation

considered and the spectral bandwidth of relevance vary. The

challenges to developing consensus are exacerbated further by

the complexity of the coupled external circuit to be studied,

the associated losses therein, and whether or not additional

performance metrics are weighted against the harvested

power (e.g. vibration control and energy harvesting [7375]).

Nonlinear harvesters may also have multiple stable solutions

for a given operating condition, making a steady-state

performance metric ambiguous at best. The practice of many

works in this review has been to directly compare tested

or simulated results of bistable power harvesters with the

linear equivalents. This approach is suitable for individual

case studies but does not provide for general conclusions to be

drawn. Thus, broadly applicable energy harvesting efficiency

metrics or protocols remain to be determined.

6. Relation to bistable damping research

As indicated, earlier studies in energy harvesting simplified

analyses by utilizing only the mechanical governing

equation [1619]. This presumes that the net mechanicalenergy dissipated would serve as a theoretical ceiling on

the harvested power and that electromechanical coupling

is equivalent to additional velocity-proportional damping.

If this perspective is maintained, contemporaneous work in

bistable vibration damping should be recognized and noted

for analytical results that do not have counterparts in existing

bistable energy harvesting research but which may be useful

for future studies.

Avramov and Mikhlin [76] considered the vibration

absorption capability of a bistable snap-through truss attached

to a main elastic system. The method of nonlinear normal

modes (NNM) was employed and found to be accurateas compared with direct numerical integration when the

snap-through oscillator exhibited intrawell oscillations. Once

snap-through occurred, trajectories predicted by NNM

diverged from simulation but relative modal amplitudes

remained consistent. It was found that localization of the

NNM could be attained within the snap-through truss,

thus maximizing vibration energy transfer to that element.

The localized NNM was shown to be stable using MMS.

Gendelman and Lamarque [77]also used MMS to determine

dynamic manifolds for a bistable oscillator and coupled host

vibrating oscillator. Three distinct zones were determined that

indicated efficient energy pumping into the bistable device,

energy dissipated via intrawell vibration, and transient chaos.Numerically integrated simulations verified the regimes and

the approximate bounds amongst the predicted manifolds.

These results may provide insight and initial direction to

coupled bistable energy harvesting research.

Bistable vibration damping studies have a variety of

protocols regarding efficiency and energy transfer [78] which

may be of benefit in the ultimate determination of energy

harvesting metrics. Johnson et al [79] applied a loss factorcriteria to a bistable snap-through device; while convergence

of the measure was shown, it was suggested that it may not

always be used when the oscillator undergoes chaos vibration.

It was also illustrated that bistable devices can be used for

designing adaptive damping with respect to input amplitude

and frequency [79]. Studies in micro- and nano-metamaterials

having bistable inclusions for increased vibration and acoustic

damping use a variety of methods for material performance

evaluation [8083]. These concepts should be considered in

the resolution of energy harvesting metrics generally, and

may provide clear, comparable evidence of the dramatic

advantages of bistable harvesters thus displayed in analyses

and experiments.

7. Concluding remarks

The benefits of exploiting bistable nonlinearities in vibration

energy harvesting have been the impetus for much recent

research. A breadth of studies have been undertaken to

shed light on the intricate electromechanical dynamics and

to provide experimental evidence of the predicted benefits.

The various bistable harvester designs thus far studied have

relied heavily on magnetic attraction, magnetic repulsion,

and mechanical loading to induce the bistability. Other

investigations have employed the bistability mechanism itselfas a novel excitation source for frequency up-conversion

applications. Depending on the excitation environment, either

the periodic excitation of bistable interwell dynamics or a

number of frequency up-conversion techniques can be utilized

to provide practical energy harvesting output, exemplifying

the versatility of bistable harvester designs.

On the whole, bistable harvesters are an improvement

upon their linear counterparts in steady-state vibration

environments and have been analytically and experimentally

shown to provide as much as an order of magnitude increase

in harvested energy. The benefits or disadvantages of bistable

devices due to stochastic excitation have not yet beenconclusively determined and a genuine need remains to

better understand the potential of vibration energy harvesting

in random excitation environments. A number of advanced

topics have only begun to be explored, such as the accurate

and reliable prediction of high-energy bistable dynamics

for maximum harvesting performance and the opportunities

provided for by multi-degree-of-freedom systems containing

bistable elements. Concentrated efforts are required to

answer the questions involved for realistic vibration energy

harvesting with bistable devices. Researchers in the field

may also find inspiration from contemporaneous work in

advanced metamaterials and bistable vibration damping. To

date, vibration energy harvesting studies have drawn uponthe expertise from members among a number of research

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communities in order to solve the problems of optimum

device development and analytical assessment. Continued

collaborative efforts will be necessary to formulate novel

solutions and implementations to the successful utilization of

bistable systems as an effective and robust energy harvesting

platform.

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